Deep learning for algorithm portfolios

ABSTRACT

Automated feature construction for algorithm portfolios in machine learning is provided. A gray scale image is generated from a text representing a problem instance. The gray scale image is rescaled or reshaped to a predefined size that is smaller than an initial size of the gray scale image. The rescaled gray scale image represents features of the problem instance. The rescaled gray scale image is input as features to a machine learning-based convolutional neural network. Based on the rescaled gray scale image, the machine learning-based convolutional neural network is automatically trained to learn to automatically determine one or more problem solvers from a portfolio of problem solvers suited for solving the problem instance.

FIELD

The present application relates generally to computers and computerapplications, and more particularly to computer-implemented machinelearning and automated feature construction for algorithm portfoliosusing computer-implemented deep learning.

BACKGROUND

In many scenarios there is no single solver that will dominate orprovide optimal performance across a wide range of problem instances.Algorithm selection in algorithm portfolios is designed to help identifythe best approach for each problem at hand, to select the best algorithmfor a given problem instance. This selection is usually based oncarefully constructed features that characterize problem instances,designed to quickly present the overall structure of the probleminstance as a constant size numeric vector. Based on these features, anumber of machine learning techniques can be utilized to predict theappropriate solver to execute, leading to improvements over relyingsolely on any one solver. However, feature generation usually involves amanually process, and the creation of good features becomes an arduoustask requiring a great deal of knowledge of the problem domain ofinterest and expertise.

BRIEF SUMMARY

A computer-implemented method and system that automates featureconstruction for algorithm portfolios in machine learning may beprovided. The method, in one aspect, may include receiving a probleminstance represented as text describing a problem to be solved bycomputer-implemented problem solver. The method may also includegenerating a gray scale image from the text by converting the text intothe gray scale image that corresponds to the text. The method may alsoinclude rescaling the gray scale image to a predefined size that issmaller than an initial size of the gray scale image, into a rescaledgray scale image, the rescaled gray scale image representing features ofthe problem instance. The method may also include inputting the rescaledgray scale image as features to a machine learning-based convolutionalneural network. The method may also include training based on therescaled gray scale image, by one or more of the processors, the machinelearning-based convolutional neural network to learn to automaticallydetermine one or more problem solvers from a portfolio of problemsolvers suited for solving the problem instance. The receiving,generating, rescaling and inputting may be performed for a plurality ofproblem instances for the training.

A system of automated feature construction for algorithm portfolios inmachine learning, in one aspect, may include one or more memory devicesand one or more hardware processors operatively coupled to the memorydevice. One or more of the hardware processors may be operable toreceive a problem instance represented as text describing a problem tobe solved by computer-implemented problem solver. One or more of thehardware processors may be further operable to generate a gray scaleimage from the text by converting the text into the gray scale imagethat corresponds to the text. One or more of the hardware processors maybe further operable to rescale the gray scale image to a predefined sizethat is smaller than an initial size of the gray scale image, into arescaled gray scale image, the rescaled gray scale image representingfeatures of the problem instance. One or more of the hardware processorsmay be further operable to train based on the rescaled gray scale imageas features, a machine learning-based convolutional neural network tolearn to automatically determine one or more problem solvers from aportfolio of problem solvers suited for solving the problem instance.One or more of the hardware processors may be operable to receivemultiple problem instances, generate multiple gray scale images andrescale to multiple rescaled gray scale images respectively, themultiple rescaled gray scale images used as a training dataset fortraining the machine learning-based convolutional neural network.

A computer readable storage medium or device storing a program ofinstructions executable by a machine to perform one or more methodsdescribed herein also may be provided.

Further features as well as the structure and operation of variousembodiments are described in detail below with reference to theaccompanying drawings. In the drawings, like reference numbers indicateidentical or functionally similar elements.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram illustrating an overview of a methodology of thepresent disclosure in one embodiment.

FIG. 2 is a diagram illustrating an overview of converting textrepresentation of a problem instance to an image representing featuresfor machine learning in one embodiment of the present disclosure.

FIGS. 3A, 3B, 3C, 3D and 3E show example problem instances converted toimages.

FIG. 4 illustrates a CNN model employed with an image representation oftextual information for performing algorithm selection in one embodimentof the present disclosure.

FIG. 5 is a flow diagram illustrating a method of the present disclosurein one embodiment.

FIG. 6 is a diagram showing components of a system in one embodiment ofthe present disclosure.

FIG. 7 illustrates a schematic of an example computer or processingsystem that may implement an automated feature extraction and neuralnetwork machine learning in one embodiment of the present disclosure.

DETAILED DESCRIPTION

A system, method and/or technique are disclosed for an automatedmethodology for producing an informative set of features utilizing adeep neural network. In one embodiment, the methodology completelyautomates an algorithm selection pipeline, achieving significantlybetter performance than using a single best solver across multipleproblem domains.

A solver refers to a computer-implemented algorithm that solves aproblem, for example, in a variety of domains. Rather than being generalpurpose programs, solvers may be specialized for a small subset ofproblems. Given the number of specialized solvers (a portfolio ofsolvers), algorithm selection techniques identify or determine which ofthe solvers should be used for a given problem instance. Such algorithmselection techniques rely on the quality of a set of structural featuresthey use to distinguish among the problem instances. If the features aretoo noisy or uninformative, a selection technique may be unable to makeintelligent decisions.

A computer-implemented autonomous and automatic approach to the featuregeneration process in one embodiment of the present disclosure mayemploy a deep learning approach, providing a new training approach. Inmany domains, the specifics of any problem instance are expressed as atext document. For example, a Satisfiability (SAT) problem is typicallyrepresented in the DIMACS conjunctive normal form (CNF) format whereafter a header, each line in the file describes the literals in eachclause. Briefly, in computer science, the Boolean Satisfiability Problem(also referred to as Propositional Satisfiability Problem,Satisfiability or SAT) is the problem of determining if there exists aninterpretation that satisfies a given Boolean formula, for example,whether a proposition statement evaluate to true. Similarly, Constraintsatisfaction problem (CSP) can be represented in the XCSP format(extended markup language (XML) representation of constraint networks)while mixed integer programs (MIPs) can be represented in linearprogramming (LP) or Mathematical Programming System (MPS) formats.

Regardless of the format, the problem instances are represented in atext file. A methodology of the present disclosure in one embodiment mayautomatically convert any such text file into an image, e.g., agrayscale square image, which can in turn be used to train a deep neuralnetwork to predict the best solver for the instance. In one embodiment,little information may be used to predict which solver or algorithm touse, for example, by shooting a photo (creating an image) of the probleminstance and using the information in the image to predict the bestsolver. The approach in one embodiment of the present disclosure worksacross multiple problem representations, each of which may have a uniquegrammar and internal organization of data, and may vary in terms oftheir size. While most machine learning approaches rely on constantsized feature vectors, the approach of the present disclosure in oneembodiment may work across multiple problem representations with varyingsizes. A methodology of the present disclosure in one embodimentrepresents a diverse class of problem domains in a finiterepresentation.

Algorithm selection is the study of choosing the most appropriate solverfor the problem at hand based on a descriptive set of features thatdescribe the instance. In one embodiment of the present disclosure, atechnique such as the cost-sensitive hierarchical clustering (CSHC) isused to build an algorithm portfolio based on features that characterizethe problem instances. CSHC bases its decision by training a forest ofrandom trees. Each tree in the forest is trained with a splittingcriterion that ensures that it divides the training data such that theinstances in each child node, maximally agree on the solver they prefer.The partitioning stops when the amount of data in a child node becomestoo small. To ensure variation of the forest, each tree is trained on arandom subset of 70% of the data and a random subset of consideredfeatures. Therefore, whenever a new instance is presented, each treevotes for the best solver based on its data.

FIG. 1 is a diagram illustrating an overview of a methodology of thepresent disclosure in one embodiment. Problem instances may berepresented as text 102. Such text representations of problem instancesmay be received, for example, by one or more hardware processors. Aproblem instance represented as text describes a problem to be solved bya computer-implemented problem solver. The text is converted to images104. For example, one or more of the processors generates a gray scaleimage from the text by converting the text into the gray scale imagethat corresponds to the text. The gray scale image is rescaled, forexample, by one or more of the processors, to a predefined size that issmaller than the gray scale image size into a rescaled gray scale image.The rescaled gray scale image represents features of the probleminstance. In this way, automatic feature extraction may be performed bya machine. The rescaled gray scale image is input as features to amachine learning-based convolutional neural network 106. The machinelearning-based convolutional neural network is trained based on therescaled gray scale image to learn to automatically determine one ormore problem solvers from a portfolio of problem solvers suited forsolving the problem instance. As shown at 102, multiple textrepresentations corresponding to multiple problem instances may bereceived. Each of the multiple text representations may be generatedinto a corresponding image and rescaled. The rescaled gray scale imagecaptures structure of the underlying problem instance.

Each of the rescaled images may be input to train the neural network106. For instance, the neural network may be trained using the imagescorresponding to the problem instances. In one embodiment, the neuralnetwork 106 is trained to output a prediction of which algorithm to usefor a given problem instance, for example, as shown at 108.

The machine learning-based convolutional neural network that is trainedmay be executed to select the problem solver suited to solving a newproblem instance (also referred to as a test instance). The trainedneural network may be used, e.g., run or executed on a hardwareprocessor, to predict a solver for a new problem instance. The neuralnetwork may output for each solver a value between 0 and 1, where 0indicates that the solver cannot solve the problem instance and 1 meansthat the solver can solve the problem instance. For example, the trainedneural network may be used to predict which solver could finish a givenproblem instance within a given time limit or timeframe. The neuralnetwork may output for each solver a value between 0 and 1, where 0indicates that the solver cannot finish the given problem instance and 1means that the solver can finish the given problem instance. The neuralnetwork may select the solver whose output obtains the highest value.

The output layer (or other layers) of the neural network 106 may be alsoused to train an automated algorithm selection tool as shown at 110. Forexample, the output layer of the neural network may be used orinterpreted as a new feature vector for another algorithm selectionmethod or tool. The output layer of a neural network corresponds to aset of values that are mapped to a prediction of the neural network. Themethodology of the present disclosure in one embodiment may use thevalues of the output layer that the prediction is based on as a featurevector. An algorithm selection can then be trained based on thosefeature vectors and make decision what algorithm to use.

The text files 102 may have different specifications. There may manyfile specifications utilized, each one specifically formulated to mostconveniently represent a particular problem. For example, Satisfiabilityinstances are typically represented in CNF format:

c SAT EXAMPLE

p cnf 3 2

1 −3 0

2 3 −1 0

In this representation a line beginning with a “c” is considered acomment, while “p” signals the problem type, number of variables andnumber of clauses. The subsequent lines list the literals belonging toeach clause, with a minus sign depicting a negation. In the exampleabove, the mathematical SAT problem represented is: (x₁ ν

x₃)Λ(x₂ νx₃ ν

x₁).

The CNF format efficiently captures all the necessary details about theSAT problem, but at the same time is unable to capture the requirementsof a full constraint satisfaction problem. The XCSP format may be abetter fit, which uses XML to first specify the domains, then thevariables, and then the relations between the variables. The examplebelow defines a problem with variables A1 and A2 that can take a value 1or 2, each of which is different.

<domains nbDomains=“1”>

<domain name=“d0” nbValues=“2”>1 . . . 2</domain>

</domains>

<variables nbVariables=“2”>

<variable name=“A1” domain=“d0”/>

<variable name=“A2” domain=“d0”/>

</variables>

<constraint name=“c0” arity=“2”

scope=“A1 A2”

reference=“global:alldifferent”/>

The SAT problem could be represented as a CSP. Also, anynondeterministic polynomial time (NP)-complete problem may be encodedinto any other NP-complete problem in polynomial time. The methodologyof the present disclosure is able to take potentially any problemdefinition and encode into a usable entity or item by a machine learningapproach like a deep neural network. The methodology of the presentdisclosure in one embodiment takes the above-presented or any otherformats and converts them to grayscale images with a predefineddimension n. The methodology of the present disclosure may also usethese images to train and test a deep neural network. In one embodiment,the deep neural network may be tested using 10-fold-cross validation. Inone embodiment, the output 108 of the network represents a scoring ofall solvers on the provided input instances. In one embodiment, theoutput may be used directly as a selection approach. In anotherembodiment, the output may be used as new features to be used byexisting selection strategies, e.g., as shown at 110.

FIG. 2 is a diagram illustrating an overview of converting textrepresentation of a problem instance to an image representing featuresfor machine learning in one embodiment of the present disclosure. Theconversion methodology of the present disclosure in one embodiment workswith input documents and output images of different sizes. For example,the vocabulary and grammar is vastly different between the CNF and XCSPformats. Hence, for example, the methodology of the present disclosurein one embodiment works without employing known structures, and worksgenerally for all formats. In one embodiment, the typical approach ofcounting the frequency of words is not considered in the methodology ofthe present disclosure. The methodology of the present disclosure in oneembodiment may consider the relations of which words they appear nextto. Problems can be of widely different sizes, for example, with SATinstances ranging from hundreds to millions of clauses and CSP instanceincluding just dozens of lines. The methodology of the presentdisclosure in one embodiment provides a way to represent those differentformats.

The conversion process in one embodiment may work as follows. Theprocess in one embodiment is performed automatically by one or morehardware processors. For a problem instance represented in text 202, theplain text file is read character by character and replaced with itscorresponding ASCII code 204, for example, retrieved from ASCII tablestored in a memory device or the like. Each such code is stored in avector 206 of length N, where N is the number of total characters in theinput file. The vector 206 may be stored on a memory device and/orstorage device. After reading the entire file, the vector is reshapedusing the new dimension √{square root over (N)}. For instance, if thevector 206 is of length 9 then the resulting in the two dimensionalvector of size 3×3. The methodology may draw a square gray scale image208 for each problem instance 202 using the ASCII code value 204 for ashade of gray.

For example, the following snippet of a CNF file:

88 1134 1972 0

699 81 −1082 0

−239 −1863 1594 0

is represented as the following vector of ASCII codes:

[56,56,32,49,49,51,52,32,49,57,55, . . . ].

In one embodiment, all characters are mapped—including spaces and linebreak symbols. ASCII codes range between 0 and 255 and they can bemapped to gray scale. In one embodiment, there exists a one-to-onemapping from the original text to the image and vice versa, andtherefore, the initial image representation is loss-free. Themethodology of the present disclosure in one embodiment rescales theimage (the initial image) to a predefined size, for example, usingstandard image scaling operators. An example of a predefined size maybe, 128×128 pixel size. Another example of a predefined size may be32×32. Yet another example may be 256×256. In one embodiment, theconversion process of the present disclosure produces a set of imageswhich are all of the same size. Instances expose structure andself-similarity (patterns can be visualized using images) and theseproperties can be maintained once the images are rescaled. The imagescan be useful to visualize and to analyze these structures, regardlessof the considered domain (or the problem instance). Images may be large,for example, 300 megabytes (MB). Reshaping reduces images to smallerdimensions, e.g., 16 kilobytes (KB). Maintaining the appearancemaintains the needed structure. While scaling the images incurs a lossin information, the retained structure is sufficient to address decisionproblems such as algorithm selection.

FIGS. 3A, 3B, 3C, 3D and 3E show example problem instances converted toimages. FIG. 3A shows an image 304 extracted (converted) from an exampleSAT problem instance that is represented in a vector of ASCII code 302.Similarly, FIG. 3B shows an image 308 extracted (converted) from anotherexample SAT problem instance that is represented in a vector of ASCIIcode 306. FIG. 3C shows an image 312 extracted (converted) from yetanother example SAT problem instance that is represented in a vector ofASCII code 306. FIG. 3D shows an image 316 extracted (converted) from aCSP instance that is represented in a vector of ASCII code 314. FIG. 3Eshows an image 320 extracted (converted) from a CSP instance that isrepresented in a vector of ASCII code 318. Patterns are visible in eachimage 304, 308, 312, 316, 320 in those figures.

In machine learning, a neural network is a structure which may be usedfor classification or regression tasks when the high dimensionality andnon-linearity of the data make these tasks hard to accomplish. In therealm of visual data, convolutional neural networks (CNN) is employed.CNNs relates to the hierarchy of the cells in visual neuroscience andits structure roughly resembles the one in the visual cortex. CNNs maybe used in image classification area and others such as speechrecognition and face recognition. Convolutional neural networks aredesigned to deal with multi-dimensions input such as images.

The modular approach of the deep network and the use of convolutionallayers allow the early stage of the network to search for junctions offeatures while the use of pooling layers try to merge them semantically.An embodiment of the present disclosure utilizes convolutional neuralnetworks in the context of algorithm portfolios: an image representationof textual information leverages the capabilities of CNNs to performalgorithm selection in one embodiment of the present disclosure.

The following describes the employed CNN model in one embodiment of thepresent disclosure. FIG. 4 illustrates a structure of a CNN modelemployed with an image representation of textual information forperforming algorithm selection in one embodiment of the presentdisclosure. An example deep CNN network of the present disclosure in oneembodiment starts with three convolutional layers 404, 406, 408, eachone followed by a max-pool layer and a dropout layer. At the very end ofthe network there are two fully connected layers 410, 412 with a singledropout layer in the middle. The output layer 414 is composed of mnodes, where m is the number of solvers. Dropout layers help to preventover fitting and make the neural network performances more stable incombination with other techniques, such as adjusting momentum andlearning rate during the training phase. Input layer 402 receives animage that represents extracted features of a problem instance (e.g.,rescaled gray scale image).

In one embodiment, the network uses the stochastic gradient descent(SGD) algorithm to speed-up the back-propagation and during the trainingphase it is updated using Nesterov momentum. In this example, theminibatch size is set to 128, learning rate is initially set to 0.03 andmomentum is initially set to 0.9. Both are adjusted during the trainingphase with step size of 0.003 for learning rate and 0.001 for momentum.The non-linearity used is the rectify function φ(x)=max(0, x), while theoutput layer uses the sigmoid function φ(x)=1/(1+e^(−x)).

FIG. 4 also shows the number of filters used and their dimensions aswell as the dimension for each convolutional layers and theprobabilities used by the dropout layers and max-pool layers. Beforetraining the neural network, the data is preprocessed in the followingway in one embodiment: For each feature, the methodology of the presentdisclosure subtracts the mean and normalizes each feature to have astandard deviation equal to 1. This preprocessing step is beneficial forefficiency and performances of neural networks. In one embodiment of thepresent disclosure, the methodology of the present disclosure trains theneural network to predict whether a solver can solve the given instanceor not, for example, as a binary regression task. The objective lossfunction is the binary cross entropy:L=−t log(p)−(1−t)log(1−p)where tε{0,1} is the ground truth value, and pε[0, 1] is the predictedvalue. In another aspect, the plain classification task, i.e., trainingthe neural network to predict which is the best solver to use for agiven instance, may be performed.

The methodology of the present disclosure in one embodiment is obliviousto any domain specific properties, for example, since it is parsingproblem instances character by character and does not rely on any givenpredefined structure. In general, the process exhibits very littlebias—except for the step that scales the initial image to its miniaturesized version. The scaling function (e.g., the default image scalingalgorithm) employed in the methodology of the present disclosure in oneembodiment retains the structure that is needed to perform algorithmselection. In one embodiment, this reduction function may be learned(e.g., by a machine) so that the needed structure is retained withoutdepending on an arbitrary transformation. For instance, the methodologymay employ, for example, a Long short-term memory (LSTM) networktechnique to learn the appropriate transformation function.

The neural network, for example, may be implemented on a hardwareprocessor, for example, using computer programming language such asPython 2.7 and Lasagne 0.1. Lasagne is a framework based on Theano 0.7which allows development of CNNs at an abstract and transparent level.In addition the framework allows exploitation of high performancegraphical processing units (GPUs). Lasagne composes the neural networkusing different available layers stacking them on top of each other. Foreach layer it is possible to alter various parameters based on thelayer's type. This leads to the implementation of the deep neuralnetwork.

As an example, training of the neural network may use a number ofepochs, for example, 100 or more epochs. The predefined image size towhich the images are scaled may be configurable (e.g., 32×32, 128×128,256×256, or another). For domains with the highest number of instances(e.g., random) larger images may provide better performances, whileimage size such as 128×128 may provide a trade-off between number ofinput parameters and performance.

FIG. 5 is a flow diagram illustrating a method of the present disclosurein one embodiment, for example, also described with reference to FIG. 1.The method automates feature construction for algorithm portfolios inmachine learning. The method in one embodiment is performed or executedby one or more hardware processors. At 502, a problem instance isreceived, which is represented as text describing a problem to be solvedby computer-implemented problem solver. At 504, a gray scale image isgenerated from the text by converting the text into the gray scale imagethat corresponds to the text. At 506, the gray scale image is rescaledto a predefined size that is smaller than an initial size of the grayscale image. For sake of description, the gray scale image that isrescaled is referred to as a rescaled gray scale image. The rescaledgray scale image represents features of the problem instance. Automaticfeature extraction hence is effected by image generation and scaling. At508, the rescaled gray scale image is input as features to a machinelearning-based convolutional neural network. At 510, based on therescaled gray scale image, the machine learning-based convolutionalneural network is trained to learn to automatically determine one ormore problem solvers from a portfolio of problem solvers suited forsolving the problem instance. The receiving, generating, rescaling andinputting are performed for a plurality of problem instances, forexample, wherein the plurality of corresponding rescaled images are usedas a dataset for the training.

At 512, the machine learning-based convolutional neural network that istrained may be executed to select the problem solver suited to solving anew problem instance. The processing at 512 describes a test phase ofthe neural network. For instance, once the machine learning-basedconvolutional neural network is trained at 510, it may be tested with anew problem instance. Similarly to steps 502, 504 and 506, the newproblem instance may be converted to an image. The converted image isinput to the machine learning-based convolutional neural network, forthe neural network to output its prediction.

FIG. 6 is a diagram showing components of a system in one embodiment ofthe present disclosure. One or more hardware processors 602 may beoperatively coupled to one or more memory devices 604. One or more ofthe hardware processors 602 may receive a problem instance representedas text 606 describing a problem to be solved by computer-implementedproblem solver. The text 606, for example, may be stored in andretrieved from a storage device 608. The storage device may be coupledor connected to one or more of the hardware processors via a computernetwork or a communication network, or directly as a local storagedevice. One or more of the hardware processors may generate a gray scaleimage from the text by converting the text into the gray scale imagethat corresponds to the text. The gray scale image may be stored on amemory device 604 and/or the storage device 608. One or more of thehardware processors may rescale the gray scale image to a predefinedsize that is smaller than an initial size of the gray scale image, intoa rescaled gray scale image, the rescaled gray scale image representingfeatures of the problem instance. The rescaled gray scale image may bestored on the memory device 604 and/or the storage device 608. One ormore of the hardware processors 602 may train, based on the rescaledgray scale image as features, a machine learning-based convolutionalneural network 610 to learn to automatically determine one or moreproblem solvers from a portfolio of problem solvers suited for solvingthe problem instance. For generating a dataset for inputting to themachine learning-based convolutional neural network 610 for its trainingor learning phase, one or more of the hardware processors receivesmultiple problem instances, generates multiple gray scale images andrescales to multiple rescaled gray scale images respectively. In oneaspect, the multiple problem instances are of different problem domains.For instance, one of the multiple problem instance may be of a differentdomain from another one of the multiple problem instance. In a testphase, one or more of the hardware processors 602 may execute themachine learning-based convolutional neural network 610 that is trained,to select the problem solver suited to solving a new problem instance.

As experimental runs, the methodology of the present disclosure isempirically evaluated using different data sets, e.g., from theSatisfiability (SAT) and constraint programming domains (CSP). The SATdatasets are usually divided into the following three sub-domains:industrial, random and crafted. The experiment includes the performancesof 29 solvers for each of about 800 instances for industrial, more than2,200 instances of random and about 730 of crafted. The experiment alsouses the performances of 22 solvers for each of the almost 1,500instances in the CSP domain. The CSP instances include non-trivialinstances from problem classes such as Timetabling, FrequencyAssignment, Job-Shop, Open-Shop, Quasi-group, Costas Array, GolombRuler, Latin Square, All Interval Series, Balanced Incomplete BlockDesign, and many others. This set includes both small and large parityconstraints and all of the global constraints used during the CSP solverruns: all-different, element, weighted sum, and cumulative.

For each dataset the experiment performed the prediction task using a10-fold cross validation approach. Hence, the experiment first splits adataset into a training and test set. The train set is then splitfurther into train and validation splits using a ratio of 75%=25%. Theneural network is trained using the images corresponding to theinstances of a given dataset. The trained neural network is then used topredict which solver could finish a given test instance within the timelimit. The neural network outputs for each solver a value between 0 and1, where 0 indicates that the solver cannot finish the given instanceand 1 means the opposite. For evaluation, the experiment selects thesolver whose output obtains the highest value. This strategy in thepresent disclosure is referred to as CNN. In another aspect, instead ofrelying solely on the neural network to make the correct decision onwhich solver to use, it is also possible to interpret the output layeras a new feature vector. A specialized approach for algorithm selectioncan then be used to try to refine the selection process. This approachin the present disclosure is referred to as “New Feat”.

The results obtained by the methods of the present disclosure in theabove-described experiments are compared with the ones obtained usingregular manually crafted features with CSHC, which represents astate-of-the-art approach in the area of algorithm portfolios. In oneembodiment, the methodology of the present disclosure implements aclassifier to use. Similar to a majority class in a plain classificationtask the baseline in this setting is the following: after executing allsolvers on the train dataset and computing the average running timeelapsed by each one, the experiment chooses as prediction the algorithmthat behaves on average the best. This selection strategy in the presentdisclosure is labeled or referred to as the Best Single Solver (BSS).

The empirical results show that the methodology of the presentdisclosure provides better performances than the baseline on allconsidered domains. The methodology of the present disclosure in oneembodiment is performed without relying on features crafted by experthumans. The performance may be boosted further if more problem instanceswould be available for training.

Using the experiment results, it is also considered how the methodologyof the present disclosure in one embodiment performs in terms of averagerun time per instance. For instance, the prediction is compared from theneural network not only in terms of number of solved instances but alsoin terms of average runtime used by the prediction to solve theinstance. The results show that the methodology of the presentdisclosure in one embodiment is performing better then the BSS in all ofthe scenarios and the gap to the state-of-the-art is within reasonablelimits for a fully automated approach.

With the experimental results, the number of misclassifications that theneural network may incur. e.g., on SAT instances may be assessed. Forinstance, the last layer of the neural network of the present disclosurein one embodiment is a vector of values in [0, 1], where 0 in position imeans that the i-th solver cannot solve the given instance, 1 otherwise.For any test instance and for any solver, the actual ability of thesolver (e.g., solve or not solve) is known. Given a test instance, howmany output values of the neural network are wrong can be counted bychanging to 0 all the values less than equal to 0.5 and changing to 1the others, comparing the outcomes with reality. The comparison showsthat the neural network makes very few errors. As long as the CNN(neural network of the present disclosure in one embodiment) selects asolver that can solve the instance the resulting portfolio will yieldgood performance. In addition existing algorithm selection techniquescan further learn patterns on top of the predictions made by the neuralnetwork of the present disclosure to automatically correct mistakes andimprove overall predictions.

The results show that the methodology of the present disclosure in oneembodiment that converts textual representations of problem instancesinto gray scale images captures structure of the underlying instance.This structure can then be picked up and exploited with the right tools.The results show that human expertise in feature generation orextraction can be removed from algorithm selection, allowing the toolsto be readily applied to new fields.

Algorithm selection is changing the practice of research of algorithms.Instead of creating methodologies that work for all, focus may be placedon highly specialized techniques, selecting the best strategy for theproblem at hand. However, in order to get selection techniques to workto their full potential a substantial amount of human expertise isrequired to create the features necessary to differentiate betweeninstances. In the present disclosure, a methodology is disclosed thatfully automates this process, which for example, may make algorithmselection an easier tool for researchers to use off-the shelf. In oneembodiment, deep learning techniques is used in automated algorithmportfolios by training a neural network using images extracted fromproblem instances. This approach provides solid performances ondifferent domains, even without using any domain knowledge. Avoidingfeature generation and the usage of domain knowledge makes this newapproach useful on a variety of different domains. The methodology ofthe present disclosure may out-perform any single solver.

In another aspect, the methodology in one embodiment may furtherintroduce domain knowledge to filter out repetitive irrelevant words.Yet in another aspect, another way of encoding words rather thancharacter by character may be possible. Still yet in another aspect, themethodology of the present disclosure may include learning how tocompress the initial image in order to retain the necessary structure,for instance, to improve performance further, and also to remove thebias induced, if any, by the employed scaling function. Still yet, themethodology may leverage existing knowledge representations such as theone available from ImageNet (image database) to improve performance.Further yet, manually crafted and automatically generated features maybe used for training the neural network, for example, by append theoriginal and new features and using the composed feature vector to trainthe neural network.

FIG. 7 illustrates a schematic of an example computer or processingsystem that may implement an automated feature extraction and neuralnetwork machine learning in one embodiment of the present disclosure.The computer system is only one example of a suitable processing systemand is not intended to suggest any limitation as to the scope of use orfunctionality of embodiments of the methodology described herein. Theprocessing system shown may be operational with numerous other generalpurpose or special purpose computing system environments orconfigurations. Examples of well-known computing systems, environments,and/or configurations that may be suitable for use with the processingsystem shown in FIG. 7 may include, but are not limited to, personalcomputer systems, server computer systems, thin clients, thick clients,handheld or laptop devices, multiprocessor systems, microprocessor-basedsystems, set top boxes, programmable consumer electronics, network PCs,minicomputer systems, mainframe computer systems, and distributed cloudcomputing environments that include any of the above systems or devices,and the like.

The computer system may be described in the general context of computersystem executable instructions, such as program modules, being executedby a computer system. Generally, program modules may include routines,programs, objects, components, logic, data structures, and so on thatperform particular tasks or implement particular abstract data types.The computer system may be practiced in distributed cloud computingenvironments where tasks are performed by remote processing devices thatare linked through a communications network. In a distributed cloudcomputing environment, program modules may be located in both local andremote computer system storage media including memory storage devices.

The components of computer system may include, but are not limited to,one or more processors or processing units 12, a system memory 16, and abus 14 that couples various system components including system memory 16to processor 12. The processor 12 may include module(s) 10 that performsthe methods described herein. The module(s) 10 may be programmed intothe integrated circuits of the processor 12, or loaded from memory 16,storage device 18, or network 24 or combinations thereof.

Bus 14 may represent one or more of any of several types of busstructures, including a memory bus or memory controller, a peripheralbus, an accelerated graphics port, and a processor or local bus usingany of a variety of bus architectures. By way of example, and notlimitation, such architectures include Industry Standard Architecture(ISA) bus, Micro Channel Architecture (MCA) bus, Enhanced ISA (EISA)bus, Video Electronics Standards Association (VESA) local bus, andPeripheral Component Interconnects (PCI) bus.

Computer system may include a variety of computer system readable media.Such media may be any available media that is accessible by computersystem, and it may include both volatile and non-volatile media,removable and non-removable media.

System memory 16 can include computer system readable media in the formof volatile memory, such as random access memory (RAM) and/or cachememory or others. Computer system may further include otherremovable/non-removable, volatile/non-volatile computer system storagemedia. By way of example only, storage system 18 can be provided forreading from and writing to a non-removable, non-volatile magnetic media(e.g., a “hard drive”). Although not shown, a magnetic disk drive forreading from and writing to a removable, non-volatile magnetic disk(e.g., a “floppy disk”), and an optical disk drive for reading from orwriting to a removable, non-volatile optical disk such as a CD-ROM,DVD-ROM or other optical media can be provided. In such instances, eachcan be connected to bus 14 by one or more data media interfaces.

Computer system may also communicate with one or more external devices26 such as a keyboard, a pointing device, a display 28, etc.; one ormore devices that enable a user to interact with computer system; and/orany devices (e.g., network card, modem, etc.) that enable computersystem to communicate with one or more other computing devices. Suchcommunication can occur via Input/Output (I/O) interfaces 20.

Still yet, computer system can communicate with one or more networks 24such as a local area network (LAN), a general wide area network (WAN),and/or a public network (e.g., the Internet) via network adapter 22. Asdepicted, network adapter 22 communicates with the other components ofcomputer system via bus 14. It should be understood that although notshown, other hardware and/or software components could be used inconjunction with computer system. Examples include, but are not limitedto: microcode, device drivers, redundant processing units, external diskdrive arrays, RAID systems, tape drives, and data archival storagesystems, etc.

The present invention may be a system, a method, and/or a computerprogram product. The computer program product may include a computerreadable storage medium (or media) having computer readable programinstructions thereon for causing a processor to carry out aspects of thepresent invention.

The computer readable storage medium can be a tangible device that canretain and store instructions for use by an instruction executiondevice. The computer readable storage medium may be, for example, but isnot limited to, an electronic storage device, a magnetic storage device,an optical storage device, an electromagnetic storage device, asemiconductor storage device, or any suitable combination of theforegoing. A non-exhaustive list of more specific examples of thecomputer readable storage medium includes the following: a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (ROM), an erasable programmable read-only memory (EPROMor Flash memory), a static random access memory (SRAM), a portablecompact disc read-only memory (CD-ROM), a digital versatile disk (DVD),a memory stick, a floppy disk, a mechanically encoded device such aspunch-cards or raised structures in a groove having instructionsrecorded thereon, and any suitable combination of the foregoing. Acomputer readable storage medium, as used herein, is not to be construedas being transitory signals per se, such as radio waves or other freelypropagating electromagnetic waves, electromagnetic waves propagatingthrough a waveguide or other transmission media (e.g., light pulsespassing through a fiber-optic cable), or electrical signals transmittedthrough a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network may comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device.

Computer readable program instructions for carrying out operations ofthe present invention may be assembler instructions,instruction-set-architecture (ISA) instructions, machine instructions,machine dependent instructions, microcode, firmware instructions,state-setting data, or either source code or object code written in anycombination of one or more programming languages, including an objectoriented programming language such as Smalltalk, C++ or the like, andconventional procedural programming languages, such as the “C”programming language or similar programming languages. The computerreadable program instructions may execute entirely on the user'scomputer, partly on the user's computer, as a stand-alone softwarepackage, partly on the user's computer and partly on a remote computeror entirely on the remote computer or server. In the latter scenario,the remote computer may be connected to the user's computer through anytype of network, including a local area network (LAN) or a wide areanetwork (WAN), or the connection may be made to an external computer(for example, through the Internet using an Internet Service Provider).In some embodiments, electronic circuitry including, for example,programmable logic circuitry, field-programmable gate arrays (FPGA), orprogrammable logic arrays (PLA) may execute the computer readableprogram instructions by utilizing state information of the computerreadable program instructions to personalize the electronic circuitry,in order to perform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions.

These computer readable program instructions may be provided to aprocessor of a general purpose computer, special purpose computer, orother programmable data processing apparatus to produce a machine, suchthat the instructions, which execute via the processor of the computeror other programmable data processing apparatus, create means forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks. These computer readable program instructionsmay also be stored in a computer readable storage medium that can directa computer, a programmable data processing apparatus, and/or otherdevices to function in a particular manner, such that the computerreadable storage medium having instructions stored therein comprises anarticle of manufacture including instructions which implement aspects ofthe function/act specified in the flowchart and/or block diagram blockor blocks.

The computer readable program instructions may also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational steps to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the block may occur out of theorder noted in the figures. For example, two blocks shown in successionmay, in fact, be executed substantially concurrently, or the blocks maysometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

The terminology used herein is for the purpose of describing particularembodiments only and is not intended to be limiting of the invention. Asused herein, the singular forms “a”, “an” and “the” are intended toinclude the plural forms as well, unless the context clearly indicatesotherwise. It will be further understood that the terms “comprises”and/or “comprising,” when used in this specification, specify thepresence of stated features, integers, steps, operations, elements,and/or components, but do not preclude the presence or addition of oneor more other features, integers, steps, operations, elements,components, and/or groups thereof.

The corresponding structures, materials, acts, and equivalents of allmeans or step plus function elements, if any, in the claims below areintended to include any structure, material, or act for performing thefunction in combination with other claimed elements as specificallyclaimed. The description of the present invention has been presented forpurposes of illustration and description, but is not intended to beexhaustive or limited to the invention in the form disclosed. Manymodifications and variations will be apparent to those of ordinary skillin the art without departing from the scope and spirit of the invention.The embodiment was chosen and described in order to best explain theprinciples of the invention and the practical application, and to enableothers of ordinary skill in the art to understand the invention forvarious embodiments with various modifications as are suited to theparticular use contemplated.

We claim:
 1. A computer-implemented method of automated featureconstruction for algorithm portfolios in machine learning, comprising:receiving, by one or more processors, a problem instance represented astext describing a problem to be solved by computer-implemented problemsolver; generating, by one or more of the processors, a gray scale imagefrom the text by converting the text into the gray scale image thatcorresponds to the text; rescaling, by one or more of the processors,the gray scale image to a predefined size that is smaller than aninitial size of the gray scale image, into a rescaled gray scale image,the rescaled gray scale image representing features of the probleminstance; inputting, by one or more of the processors, the rescaled grayscale image as features to a machine learning-based convolutional neuralnetwork; and training based on the rescaled gray scale image, by one ormore of the processors, the machine learning-based convolutional neuralnetwork to learn to automatically determine one or more problem solversfrom a portfolio of problem solvers suited for solving the probleminstance, wherein the receiving, generating, rescaling and inputting areperformed for a plurality of problem instances for the training.
 2. Themethod of claim 1, further comprising: executing, by one or more of theprocessors, the machine learning-based convolutional neural network thatis trained, to select the problem solver suited to solving a new probleminstance.
 3. The method of claim 1, wherein the generating comprisesreplacing each character in the text by a corresponding ASCII code andstoring the corresponding ASCII code in a vector of length N, Nrepresenting character length of the text.
 4. The method of claim 3,wherein the rescaling comprises reshaping the vector into a newdimension comprising a square root of N.
 5. The method of claim 1,wherein the rescaling comprises reshaping the gray scale image to asquare gray scale image.
 6. The method of claim 5, wherein thepredefined size is 128×128 pixels.
 7. The method of claim 1, wherein themachine learning-based convolutional neural network outputs for a solvera value between 0 and 1, wherein 0 indicates that the solver cannotfinish solving the problem instance in a given timeframe and 1represents that the solver can finish solving the problem instance inthe given timeframe.
 8. The method of claim 1, wherein an output of themachine learning-based convolutional neural network is input to analgorithm selection tool as a feature vector.
 9. A system of automatedfeature construction for algorithm portfolios in machine learning,comprising: one or more memory devices; one or more hardware processorsoperatively coupled to the memory device, one or more of the hardwareprocessors operable to receive a problem instance represented as textdescribing a problem to be solved by computer-implemented problemsolver, one or more of the hardware processors further operable togenerate a gray scale image from the text by converting the text intothe gray scale image that corresponds to the text, one or more of thehardware processors further operable to rescale the gray scale image toa predefined size that is smaller than an initial size of the gray scaleimage, into a rescaled gray scale image, the rescaled gray scale imagerepresenting features of the problem instance, one or more of thehardware processors operable to train based on the rescaled gray scaleimage as features, a machine learning-based convolutional neural networkto learn to automatically determine one or more problem solvers from aportfolio of problem solvers suited for solving the problem instance,wherein the one or more of the hardware processors receives multipleproblem instances, generates multiple gray scale images and rescales tomultiple rescaled gray scale images respectively, the multiple rescaledgray scale images used as a training dataset for training the machinelearning-based convolutional neural network.
 10. The system of claim 9,wherein the multiple problem instances are of different problem domains.11. The system of claim 9, wherein one or more of the hardwareprocessors executes the machine learning-based convolutional neuralnetwork that is trained, to select the problem solver suited to solvinga new problem instance.
 12. The system of claim 9, wherein one or moreof the hardware processors generates the gray scale image by replacingeach character in the text by a corresponding ASCII code and storing thecorresponding ASCII code in a vector of length N on the memory device, Nrepresenting character length of the text.
 13. The system of claim 12,wherein one or more of the hardware processors rescales by reshaping thevector into a new dimension comprising a square root of N.
 14. Thesystem of claim 9, wherein one or more of the hardware processorsrescales by reshaping the gray scale image to a square gray scale image.15. The system of claim 9, wherein the predefined size is 128×128pixels.
 16. The system of claim 9, wherein the machine learning-basedconvolutional neural network outputs for a solver a value between 0 and1, where 0 indicates that the solver cannot finish solving the probleminstance in a given timeframe and 1 represents that the solver canfinish solving the problem instance in the given timeframe.
 17. Thesystem of claim 9, wherein an output of the machine learning-basedconvolutional neural network is input to an algorithm selection tool asa feature vector.
 18. A computer readable storage medium storing aprogram of instructions executable by a machine to perform a method ofautomated feature construction for algorithm portfolios in machinelearning, the method comprising: receiving a problem instancerepresented as text describing a problem to be solved bycomputer-implemented problem solver; generating a gray scale image fromthe text by converting the text into the gray scale image thatcorresponds to the text; rescaling the gray scale image to a predefinedsize that is smaller than an initial size of the gray scale image, intoa rescaled gray scale image, the rescaled gray scale image representingfeatures of the problem instance; inputting the rescaled gray scaleimage as features to a machine learning-based convolutional neuralnetwork; and training based on the rescaled gray scale image the machinelearning-based convolutional neural network to learn to automaticallydetermine one or more problem solvers from a portfolio of problemsolvers suited for solving the problem instance, wherein the receiving,generating, rescaling and inputting are performed for a plurality ofproblem instances for the training.
 19. The computer readable storagemedium of claim 18, further comprising: executing the machinelearning-based convolutional neural network that is trained, to selectthe problem solver suited to solving a new problem instance.
 20. Thecomputer readable storage medium of claim 18, wherein the generatingcomprises replacing each character in the text by a corresponding ASCIIcode and storing the corresponding ASCII code in a vector of length N, Nrepresenting character length of the text and the rescaling comprisesreshaping the vector into a new dimension comprising a square root of N.